Another month, another rationality quotes thread. The rules are:
Please post all quotes separately, so that they can be upvoted or downvoted separately. (If they are strongly related, reply to your own comments. If strongly ordered, then go ahead and post them together.)
Do not quote yourself.
Do not quote from Less Wrong itself, HPMoR, Eliezer Yudkowsky, or Robin Hanson. If you'd like to revive an old quote from one of those sources, please do so here.
No more than 5 quotes per person per monthly thread, please.
Provide sufficient information (URL, title, date, page number, etc.) to enable a reader to find the place where you read the quote, or its original source if available. Do not quote with only a name.
Beyond all doubt sounds fairly dogmatic, no?
Godel proved in 1931 that Hilbert's program for a solid mathematical foundation (circa 1900) was impossible.
Beyond all doubt sounds fairly dogmatic, no? Godel proved in 1931 that Hilbert's program for a solid mathematical foundation (circa 1900) was impossible.
While I don't quite agree with your claim about what Gödel accomplished, 'beyond all doubt' is an overstatement. The history of mathematics provides many examples of apparent proofs accepted by the profession later being rejected for containing devastating errors. Even a single instance of this occurring would, strictly speaking, rule out a literal 'beyond all doubt' claim.
Well no, Goedel proved that nonstandard models of the natural numbers exist. Chaitin went on to prove that any formal axiomatic system, only containing a finite amount of axioms, will eventually face true theorems it cannot prove, for lack of information in its axioms.
That doesn't mean proven mathematical theorems are actually wrong, and unfortunately, Goedel's Platonism has resulted in most of society thinking about mathematics and proof in the wrong way.
7ike
Not everything can be proven, but those that are are proven beyond (virtually all) doubt.
Another month, another rationality quotes thread. The rules are: